Questions: On a particular day, a restaurant that is open for lunch and dinner had 179 customers. Each customer came in for one meal. An employee recorded at which meal each customer came in and whether the customer ordered dessert. The data are summarized in the table below. Dessert No dessert ----------------------------- Lunch 11 29 Dinner 45 94 Suppose a customer from that day is chosen at random. Answer each part. Do not round intermediate computations, and round your answers to the nearest hundredth. (If necessary, consult a list of formulas.) (a) What is the probability that the customer came for dinner? (b) What is the probability that the customer came for dinner or ordered dessert?

Solution Steps

To find the probability that a randomly chosen customer came for dinner, we first determine the total number of customers who came for dinner. The total number of dinner customers is given by:

\[ \text{Total Dinner Customers} = \text{Dinner Dessert} + \text{Dinner No Dessert} = 45 + 94 = 139 \]

Next, we calculate the probability:

\[ P(\text{Dinner}) = \frac{\text{Total Dinner Customers}}{\text{Total Customers}} = \frac{139}{179} \approx 0.7765 \]

Rounding to the nearest hundredth, we have:

\[ P(\text{Dinner}) \approx 0.78 \]

To find the probability that a customer either came for dinner or ordered dessert, we first calculate the total number of customers who ordered dessert:

\[ \text{Total Dessert Customers} = \text{Lunch Dessert} + \text{Dinner Dessert} = 11 + 45 = 56 \]

Now, we can calculate the individual probabilities:

\[ P(\text{Dinner}) \approx 0.78 \quad \text{(from Step 1)} \] \[ P(\text{Dessert}) = \frac{\text{Total Dessert Customers}}{\text{Total Customers}} = \frac{56}{179} \approx 0.3123 \]

Next, we find the probability of both events occurring (i.e., customers who came for dinner and ordered dessert):

\[ P(\text{Dinner and Dessert}) = \frac{\text{Dinner Dessert}}{\text{Total Customers}} = \frac{45}{179} \approx 0.2514 \]

Using the formula for the probability of the union of two events:

\[ P(\text{Dinner or Dessert}) = P(\text{Dinner}) + P(\text{Dessert}) - P(\text{Dinner and Dessert}) \]

Substituting the values:

\[ P(\text{Dinner or Dessert}) \approx 0.78 + 0.3123 - 0.2514 \approx 0.8409 \]

Rounding to the nearest hundredth, we have:

\[ P(\text{Dinner or Dessert}) \approx 0.84 \]

Final Answer